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I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.
I cross-linked Cantor space from (newly created) Examples-sections at topological space and locale and topology - contents
Eventually hopefully this sidebar for topology is expanded to something that reflects the scope of the relevant nLab articles
I added a bit more to Cantor space, including the abstract characterization up to homeomorphism (which was oddly missing, since what was there seemed to be leading right up to that point). While I was at it, I created perfect space (with perfect set redirected to it).
I gave Cantor space more of an Idea-section. Then I expanded the discussion at As a subspace of the real line with full detail. The same discussion I also copied over to Tychonoff product in this example.
I’m interested in the last sentence of the Idea section. In what way is Cantor space used to construct the Peano curve?
Worth adding the coalgebraic description? What is it, the terminal coalgebra for the endofunctor on $Top$, $X \mapsto X + X$?
Todd, sorry, I should have been more explicit. Maybe I should write “may be used to neatly organize the construction” rather than “may be used to construct”: I am thinking of picking a continuous surjection $Cantor \overset{t}{\to} Cantor \times Cantor$ (e.g. unshuffle), then observing that there is easily a continuous surjection $s \colon Cantor \to [0,1]$ and, with a tad more work, that every continuous function from $Cantor$ to a linear space may be extended along the defining embedding $Cantor \hookrightarrow [0,1]$ (by linear interpolation). Then applying this extension to the surjection $Cantor \to Cantor \overset{t}{\times} Cantor \overset{(s,s)}{\to} [0,1] \times [0,1]$ gives the desired continuous surjection.
Well, I’ll be. That’s rather nice, Urs. Never saw that before (and see nothing wrong with it).
David: that’s right.
Ok, I’ll put it in.
Being much taken with the simplicity of this Peano curve as sketched by Urs #7, I looked around and saw this is called the Lebesgue space-filling curve, which has another nice properties such as being differentiable almost everywhere. It’s obviously similar in flavor to the Cantor-Lebesgue function.
Anyway, I went ahead and bashed out the construction at Peano curve, with a proof of continuity. It was just a quick and dirty job, which I may see about cleaning up later. Not many cross-links were inserted. (It’s now bedtime for me.)
Thanks, Todd! That’s very nice. I didn’t know that Lebesgue’s name is associated with this.
I did some more jiggering with Peano curve, which then led me to add to Cantor space a proof of the Hausdorff-Alexandroff theorem, which says that every compact metric space is a continuous image of Cantor space.
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